Download Sequence of Integers Generated by Summing the Digits of their

Indian Journal of Science and Technology, Vol 8(15), DOI: 10.17485/ijst/2015/v8i15/69912, July 2015
ISSN (Print) : 0974-6846
ISSN (Online) : 0974-5645
Sequence of Integers Generated by Summing the
Digits of their Squares
H. I. Okagbue1*, M. O. Adamu2, S. A. Iyase1 and A. A. Opanuga1
Department of Mathematical Sciences, Covenant University, Canaanland, Ota, Nigeria;
[email protected]
2
Department of Mathematics, University of Lagos, Akoka, Lagos, Nigeria
1
Abstract
Objectives: To establish some properties of sequence of numbers generated by summing the digits of their respective
squares. Methods/Analysis: Two distinct sequences were obtained, one is obtained from summing the digits of squared
integers and the other is a sequence of numbers can never be obtained be obtained when integers are squared. Also
some mathematical operations were applied to obtain some subsequences. The relationship between the sequences was
­established by using correlation, regression and analysis of variance. Findings: Multiples of 3 were found to have multiples
of 9 even at higher powers when they are squared and their digits are summed up. Other forms are patternless, sequences
notwithstanding. The additive, divisibility, multiplicative and uniqueness properties of the two sequences yielded some
unique subsequences. The closed forms and the convergence of the ratio of the sequences were obtained. Strong positive
correlation exists between the two sequences as they can be used to predict each other. Analysis of variance showed that
the two sequences are from the same distribution. Conclusion/Improvement: The sequence generated by summing the
digits of squared integers can be known as Covenant numbers. More research is needed to discover more properties of the
sequences.
Keywords: Digits, Factors, Multiples, Sequence of Integers, Squares, Subsequence
1. Introduction
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144...(A).
This sequence is called the square number (integers)
which can be found on the online encyclopedia of integer
sequence A000290 – OEIS.
Also the sum of square;
0, 5, 14, 30, 55, 91, 140...(B) can be found in A000330–
OEIS.
Equation (B) is obtained from (A) and some few
examples are as follows: 5 = 1 + 4, 14 = 5 + 9, 30=14+16
The square number is the square of number (in this
case an integer) and is the outcome when an integer is
multiplied with itself1.
Many authors have worked on the square number
but this paper introduces a new concept/property of the
square number (integers) by investigating and examines
the phenomenon of summing up the digits of squared
*Author for correspondence
numbers. Weissten2 enumerated some characteristics of
the square number while some theoretical aspects can
be found in3. Some other literatures about the square
­numbers are as follows: Consecutive integers with equal
sum of squares4.
Mixed sum of Squares and Triangular Numbers5–8.
The Sum of digits of some Sequence9, 10.
The sum of Digits function of Squares11,12.
Reducing a set of subtracting squares13.
Squares of primes14.
Sequences of squares with constant second differences15.
Relationship between sequences and polynomials16.
Square free numbers17.
The sum of squares and some sequences18.
Number sequences have been applied in real life in
modeling, simulation and development of algorithms of
some carefully studied phenomena19,20.
Sequence of Integers Generated by Summing the Digits of their Squares
All these and more contributions too numerous to
mention had yielded a well-documented characteristics
of square numbers (integers) as follows;
Table 1. The first ten terms, their square
and digits sum
Number
Square
Sum of the Digits of
the Squared Number
1
1
1
2
4
4
3
9
9
4
16
7
5
25
7
6
36
9
7
49
13
8
64
10
9
81
9
10
100
1
• It is non-negative. x < 0 : x2 > 0 and x ≥ 0 : x2 ≥0.
• It increases as the integers increases.
• The ratio of two square integers is also a square.
2
2
4  2  25  5  81  9 
, =
,
=
=
9  3  16  4  100  10 
2
• A square number is also the sum of two consecutive
triangular numbers21,22.
• Square number has an odd number of positive
­divisors23. Square Divisors Number
Square
Divisors
Number
9, 4
4, 2, 1
3
9
9, 3, 1
3
16
16, 8, 4, 2, 1
5
36
36, 18, 12, 9, 6, 4, 3, 2, 1
9
etc.
• A square number cannot be a perfect number23.
4 The proper divisors of 4 are 2 and 1
2+1≠4
9 The proper divisors of 9 are 3, 2 and 1
3+2+1≠9
16 The proper divisors of 16 are 8, 4, 2 and 1
8 + 4 + 2 + 1 ≠ 16
• The only non-trivial Square Fibonacci number is
122 = 144.
2. Methodology
The first 3000 integers are squared and their respective
­digits summed up. The first 10 numbers, their square
and the sum of their respective digits are summarized in
Table 1.
3. Findings
Since the first 3000 integers are used, it was observed that
a sequence of numbers is obtained and can be grouped
in two distinct ways. First, when an integer is squared,
and the digits summed, the following numbers can be
obtained at varying frequencies which form the following
2
Vol 8 (15) | July 2015 | www.indjst.org
sequence; 1, 4, 7, 9, 10, 13, 16, 18, 19, 22, 25, 27, 28, 31, 34,
36, 37, 40, ... (C). Second, when an integer is squared and
the digits summed up, the following numbers cannot be
obtained which forms the second sequence; 2, 3, 5, 6, 8,
11, 12, 14, 15, 17, 20, 21, 23, 24, ... (D).
3.1 The Patternless Nature of the Sequences
of Odd and Even Integers when the
Digits of their Squares are Summed
Table 2 shows the results when the numbers are divided
into two distinct equivalence classes of the odd and even
integers.
There is no significance pattern of sequence formed by
each class except the multiples of 3. “Hence we state that an
even integer when squared and its digits summed yields even
or odd integer and the same applies to any odd integer”.
Table 2. The sum of the digits for odd and even
numbers
Odd
Number
Sum of the Digits of
Even
the Squared Number Number
Sum of the Digits of
the Squared Number
401
16 402
18
403
22
404
19
405
18
406
28
407
31
408
27
409
25
410
16
411
27
412
31
413
28
414
27
415
19
416
22
417
36
418
25
Indian Journal of Science and Technology
H. I. Okagbue, M. O. Adamu, S. A. Iyase and A. A. Opanuga
3.1.1 Multiples of 3
3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36 ... (E).
Table 3 shows some integers multiples of 3, their
square and their respective sum of digits: “Hence we state that any integer divisible by 3, if squared
and its digits summed yields an integer divisible by 9”.
3.1.2 Higher Powers of Multiples of 3
Even at higher powers of the multiples of 3, the same
result is obtained as shown in Table 4.
3.2 Characteristics of the Two Sequences
(C) and (D)
• (C) » (D) » 0 = ☐
• From Fibonacci sequence; A000045 – OEIS. 1, 1, 2, 3, As seen from the chart, the Lucas numbers increases more rapidly than t
Fibonacci numbers.
5, 8, 13, 21, 34, 55 ... (F)
Figure 1. Component bar chart of the first 40 Fibonacci
As seenand
from
the number.
chart, the Lucas numbers increases more rapidly than the
Lucas
Sequence (C) contains 1, 1, 13, 34, 55, ... (FA)
Fibonacci numbers.
Sequence (D) contains 2, 3, 5, 8, 21, 89 ... (FB)
1 1 10 100
3.2.1 Subsequences of Sequence C
4 2 11 20
• From Lucas sequence; A000032 – OEIS 2, 1, 3, 4, 7, 11,
5 32 40of49sequence
50
7 4numbers
Each of the
C also forms a sequence. For example, the
18, 29, 47, 76, 123, 199, 322, ... (G)
3
6
9
12
15
18
21
30
39based
45 48on51the
60numbers
90
9
100 natural numbers can be grouped
in sequence C.
Table 3. Some integers multiples of 3
Number
Square
Sum of the
Digits of Square
3
9
9
21
441
9
24
576
18
57
3249
18
63
3969
27
447
199809
36
Table 4. Higher powers of multiple of 3 and their
sums of digits.
X
x3
sum of
digits
x4
sum of
digits
x5
sum of
digits
3
27
9
81
9
243
9
6
216
9
1296
18
7776
27
9
729
18
6561
18
59049
27
12 1728
18
20736
18
248832
27
15 3375
18
50625
18
759375
36
18 5832
18
104976
27
1889568
45
Vol 8 (15) | July 2015 | www.indjst.org
10 8 19 35 46 55 71 80
16 25 29 34 38 47 52 56 61 65 70 79
1 107100
1 13
2 1113
2014 22 23 31 41 58 59 68 85 95
4 16
27 49
33 50
36 42 54 57 66 69 72 75 78 81 84 96 99
4 5 24
32 40
7 18
17
26
28
19
3053
3962
4564
48 73
51 82
60 89
90 91 98
9 3 6 9 12 15 18372144
8 1943
3574
4688
5592
7197
80
10 22
7 1667
2576
2977
3486
3894
47 52 56 61 65 70 79
13 25
6322
8723
9331 41 58 59 68 85 95
13 14
16 27
24 27 33 36 42 54 57 66 69 72 75 78 81 84 96 99
18 28
17 26
8328 37 44 53 62 64 73 82 89 91 98
19 31
43 74 88 92 97
22 34
67 76 77 86 94
25 36
87The
93 first 100 numbers and their digits sum grouped in sequenc
27 632.
Figure
28 Figure 2. The first 100 numbers and their digits sum
in sequence C.
83
31 grouped
Sequence (C) contains 1, 4, 7, 18, 76, ... (GA)
Sequence (D) contains 2, 3, 11, 29, 47, ... (GB)
• The first 40 numbers of both Fibonacci and Lucas
sequences were squared, the sum of their respective
individual numbers were obtained and the results are
represented in a component bar chart.
As seen from the chart, the Lucas numbers increases
more rapidly than the Fibonacci numbers.
Indian Journal of Science and Technology
3
yield a number in the sequence.
x 3. Multiplication of two numbers of sequence (D) yield no patte
Sequence of Integers Generated by Summing the Digits of their Squares patterned triangle similar to Paschal can be obtained which con
numbers of sequence (C) in unique arrangement.
3.2.1 Subsequences of Sequence C
4
6
Each of the numbers of sequence C also forms a sequence.
For example, the first 100 natural numbers can be grouped
based on the numbers in sequence C.
10
12
24
3.2.2 Additive Properties
• Addition of two numbers of sequence (C) can yield
numbers in both sequences (C) and (D).
• Addition of two numbers of sequence (C) can produce
numbers in the same sequence if;
(a) A multiple of 9 is added to any numbers ofsequence
(C).
(b) A multiple of 9 are added to each other.
• A pattern can be formed from the addition of the
­numbers of sequence which can be seen from
Table 1.
• Addition of two numbers of sequence (D) yield no
­pattern but a patterned triangle similar to Paschal
can be obtained which contained some numbers of
sequence (C) in unique arrangement.
3.2.3 Multiplicative Properties
• The multiplication of any two numbers of sequence
(C) yield a number in the same sequence.
Table 5. Addition of terms of sequence C.
1
4
7
9
10
13
16
18
4
5
8
11
13
14
17
20
22
7
8
11
14
16
17
20
23
25
9
10
13
16
18
19
22
25
27
10
11
14
17
19
20
23
26
28
Addition
1
15
16
18
24
22
6
9
25
30
33
40
10
15
12
18
30
36
16
24
40
22
33
24
Figure 4. Binomial table formed from multiplication of
terms in
Figure
4.sequence
BinomialD.table formed from multiplication of terms in sequ
• The multiplication of any two numbers of sequence (D)
3.2.4does
Divisibility
Properties
not necessarily
yield a number in the sequence.
• 3. Multiplication
of
two
numbers of sequence (D) yield
x Every 4th number of the sequence (C) is a multiple of 9.
no pattern but a patterned triangle similar to Paschal
obtained
contained
some
of (C).
x can
Asbe
expected
allwhich
the square
numbers
are numbers
in sequence
sequence (C) in unique arrangement.
3.2.4 Divisibility Properties
• Every 4th number of the sequence (C) is a multiple
of 9.
• As expected all the square numbers are in sequence
(C).
• All three consecutive numbers of sequence (C) are
coprime but not pairwise gcd (a, b, c ) = 1 a, b, c ∈C.
gcd (a, b ) = gcd (a, c ) = gcd (b, c ) ≠ 1.
• All four consecutive numbers of sequence (C) are
coprime but not pairwise gcd (a, b, c, d ) = 1 a, b, c, d ∈C.
gcd (a, b ) = gcd (a, c ) = gcd (a, d ) = gcd (b, c ) = gcd (b, d ) = gcd (c, d ) ≠ 1
gcd2(a, b ) 5= gcd 8(a, c )10
= gcd11
= gcd17
(a, d ) 14
(b, c ) =19gcd (b, d ) = gcd (c, d ) ≠ 1.
.
3.3 Uniqueness of Sequences C and D
Sequences (C) and (D) are unique. The complete
­respective sequences cannot be obtained by increment or
13
14
17
20
22
23
26
29 31
decrement of the numbers in the sequences rather various
16
17
20
23
25
26
29
32 34
sequences is obtained. When 1 is added to all the numbers in sequence (C), we obtain; 2, 5, 8, 10, 11, 14, 17,
18
19 22 25 27 28 31
34 36
19, 20, 23, 26, 28, 29, 32, 35, 37, 38, 41, 44... (H). When 2
is added to all numbers in sequence (C), we obtain; 3, 6,
4
9, 11, 12, 15, 18, 20, 21, 24, 27, 29, 30, 33, 36, 38, 39, 42,
5
5
45 ... (HA). When 3 is added is added to all numbers in
6
7
7
8
8
8
8
sequence (C), we obtain; 4, 7, 10, 12, 13, 16, 19, 21, 22,
10
9
10
9
10
25, 28, 30, 31, 34, 37, 39, 40, 43, 46, ... (HB). Here it can
11
11
11
11
13
13
be seen that sequence (HB) is closely related to sequence
14
14
12
14
14
13
13
(C). When 1 is subtracted from all numbers in sequence
(C), we obtain; 0, 3, 6, 8, 9, 12, 15, 17, 18, 21, 24, 26, 27, 30,
Figure 3. Binomial table obtained from addition of terms
33, 35, 36,
in sequence
D.
Figure
3. Binomial
table obtained from addition of terms in sequence
D. 39, 42, ... (HC). When 2 is subtracted from all
3.2.3 Multiplicative Properties
4
Vol 8 (15) | July 2015 | www.indjst.org
x The multiplication of any two numbers of sequence (C) yield a number in
the same sequence.
Indian Journal of Science and Technology
H. I. Okagbue, M. O. Adamu, S. A. Iyase and A. A. Opanuga
numbers in sequence (C), we obtain; –1, 2, 5, 7, 8, 11, 14,
16, 17, 20, 23, 25, 26, 29, 32, 35, 38, 41, ... (HD). When 3 is
subtracted from all numbers in sequence (C), we obtain;
–2, 1, 4, 6, 7, 10, 13, 15, 16, 19, 22, 24, 25, 28, 31, 33, 34,
37, 40, ... (HE) Here it can be seen that sequence (HE) is
closely related to sequence (C). When 1 is added to all the
numbers in sequence (D), we obtain; 3, 4, 6, 7, 9, 12, 13,
15, 16, 18, 21, 22, 24, 25, 27, 30, 31, 33, 34, ... (I). When 2 is
added to all the numbers in sequence (D), we obtain; 4, 5,
7, 8, 10, 13, 14, 16, 17, 19, 22, 23, 25, 26, 28, 31, 32, 34, 35,
... (IA). When 3 is added to all the numbers in sequence
(D), we obtain; 5, 6, 8, 9, 11, 14, 15, 17, 18, 20, 23, 24,
26, 27, 29, 32, 33, 35, 36, ... (IB). Here it can be seen that
sequence (IB) is closely related to sequence (D). When 1
is subtracted from all the numbers in sequence (D), we
obtain; 1, 2, 4, 5, 7, 10, 11, 13, 14, 16, 19, 20, 22, 23, 25,
28, 29, 31, 32, ... (IC). When 2 is subtracted from all the
numbers in sequence (D), we obtain; 0, 1, 3, 4, 6, 9, 10, 12,
13, 15, 18, 19, 21, 22, 24, 27, 28, 30, 31, ... (ID). When 3
are subtracted from all the numbers in sequence (D), we
obtain; –1, 0, 2, 3, 5, 8, 9, 11, 12, 14, 17, 18, 20, 21, 23, 26,
27, 29, 30, ... (IE). Here it can be seen that sequence (IE) is
closely related to sequence (D).
3.4.2 The Ratio of Sequence (D)
The ratio of the two successive integers of sequence (C) is
as follows: 3 , 5 , 6 , 8 , 11 , 12 , 14 , ... (K)
2 3 5 6 8 11 12
The sequence converges to almost one with a mean of
1.101494025. The closed form solution of the ratio can be
681  1 + 5 
written as: j =

 ≈ 1.1019.
250  8 
3.5 The Sequences Obtained from the
Various Factors of Sequences (C) and (D)
The first 40 members of sequences C and D are listed.
Some subsequences are obtained by the various factors
such as 2n, 3n, 4n...
3.5.1 Factors of 2
Subsequence is formed for both sequences C and D if
they are arranged based on the factors of two. The first is
3.4 The Ratio of Sequences (C) and (D)
3.4.1 The Ratio of Sequence (C)
The ratio of the two successive integers of sequence (C) is
as follows: 4 , 7 , 9 , 10 , 13 , 16 , 18 , ... (J)
1 4 7 9 10 13 16
The sequence converges to almost one with a mean of
1.11200275. The closed form solution of the ratio can be
687  1 + 5 
written as: j =

 ≈ 1.112
250  8 
Figure 6. The ratio of sequence (D). x axis - terms in
sequence D; y axis - the ratio of 2 consecutive terms of
sequence D.
Table 6. The first 10 terms of sequences C and D
n
1
2
3
4
5
6
7
8
9
10
C
1
4
7
9
10
13
16
18
19
22
D
2
3
5
6
8
11
12
14
15
17
Table 7. The 11th to 20th terms of sequences C and D
n
11
12
13
14
15
16
17
18
19
20
C
25
27
28
31
34
36
37
40
43
45
D
20
21
23
24
26
29
30
32
33
35
Table 8. The 21st to 30th terms of sequences C and D
Figure 5. The ratio of sequence (C). x axis - terms in
sequence C; y axis - the ratio of 2 consecutive terms of
sequence C.
Vol 8 (15) | July 2015 | www.indjst.org
n
21
22
23
24
25
26
27
28
29
30
C
46
49
52
54
55
58
61
63
64
67
D
38
39
41
42
44
47
48
50
51
53
Indian Journal of Science and Technology
5
Sequence of Integers Generated by Summing the Digits of their Squares
Table 9. The 31st to 40th terms of sequences C and D
n
31
C
70
72
73
D
56
57
59
32
33
34
35
36
37
38
39
40
76
79
81
82
85
88
90
60
62
65
66
68
69
71
for sequence C and the second is for sequence D. 4, 9, 13,
18, 22, 27, 31, 36, 40, 45, ... (L) 3, 6, 11, 14, 17, 21, 24, 29,
32, 35, ... (M).
3.5.2 Factors of 3
7 ,13, 19, 27, 34, 40, 46, 54, 61, 67, ... (N) 5, 11, 15, 21, 26,
32, 38, 42, 48, 53, ... (O).
3.5.3 Factors of 4
9, 18, 27, 36, 45, 54, 63, 72, 81, 90, ...(P) 6, 14, 21, 29, 35,
42, 50, 57, 65, 71, ... (Q).
3.6 The Square of Sequences C and D
New sequences are obtained from the square of sequences
C and D.
3.6.1 The Square of Sequence C
1, 16, 49, 81, 100, 169, 324, 361, 484, ...(R).
3.6.2 The Square of Sequence D
3.9 Regression Analysis of the First 40
Terms of Sequences C and D
Since there is a strong positive correlation between the
two sequences, the predictive capability of the sequences
with respect to each other is analyzed using the regression
for the first 40 terms of both sequences.
3.9.1 Sequence C as the Dependent Variable
The results of regression analysis of the two sequences
when sequence C is the dependent variable and sequence
D as the independent variable are summarized as
­follows;
The R, adjusted R square, R square and R square
change have the same value of 0.999. The regression equation is; C = 0.201 + 0.999D (1). The result of the analysis of
variance is summarized in Table 10.
3.9.2 Sequence D as the Dependent Variable
The results of regression analysis of the two sequences
when sequence D is the dependent variable and sequence
C as the independent variable are summarized as follows;
The R, adjusted R square, R square and R square change
have the same value of 0.999. The regression equation
is; D = –0.123 + 0.004C (2). The result of the analysis of
­variance is summarized in Table 11.
4, 9, 25, 36, 64, 121, 144, 196, 225, 289, ...(S).
3.10 Test of Equality of Means
3.7 The Ratio of the Sequences
The sequences have the same mean effect as summarized
in Table 12.
The ratio of the two sequences also produced some
sequences.
Table 10. ANOVA Table 1
ANOVAa,c
3.7.1 The Sequence C/D
1 4 7 9 10 13 16 , ... (T).
, , , , , ,
2 3 5 6 8 11 12
3.7.2 The Sequence D/C
2 3 5 6 8 11 12 , ... (U)
, , , , , ,
1 4 7 9 10 13 16
Model
Regression
1
6
Vol 8 (15) | July 2015 | www.indjst.org
27069.771
1
27069.771
Residual
27.729
38
.730
Total
27097.500
39
F
Sig.
37096.437 .000b
a. Dependent Variable: C b Predictors; (Constant), D.
Table 11. ANOVA Table 2
ANOVAa
3.8 Linear Correlation between Sequences
C and D
There is a strong positive correlation between the two
sequences. Pearson correlation coefficient is 0.999,
Spearman rho is 1.0 and Kendall’s tau is 1.0.
Sum of Squares Df Mean Square
Model
Regression
1
Sum of Squares Df Mean Square
17174.807
1
17174.807
Residual
17.593
38
.463
Total
17192.400
39
F
Sig.
37096.437 .000b
a. Dependent Variable: D b. Predictors: (Constant), C.
Indian Journal of Science and Technology
H. I. Okagbue, M. O. Adamu, S. A. Iyase and A. A. Opanuga
Table 12. ANOVA table for the test of equality of
means of C and D.
Source of Sum of Degrees of Mean
Variation Squares Freedom Square
Fcalculated
Ftabulated
Between 1786.05
Groups
1
1786.05 3.145455 3.963472
Within
Groups
44289.9
78
567.8192
46075.95
79
4. Conclusion
The paper have described the properties of sum of the digits of square numbers and their associated sequences and
multiples of 3 were found to be the only class of integers
with unique pattern when their digits of their square are
summed. The closed form of the ratios gave approximate
ratios. More research is needed to produce more features
and properties of the sequences. The authors proposed
that sequence C be named COVENANT NUMBERS
and be included in the online encyclopedia of integer
sequences database.
5. References
1. Conway JH, Guy RK. The Books of Numbers. 1st ed. NY:
Springer-Verlag; 1996. ISBN: 0-387-97993-X.
2. Weisstein EW. Square Number. MathWorld. 2002.
3. Ribenboim P. My Numbers, My Friends. NY: ­Springer-Verlag;
2000.
4. Alfred U. n and n+1 consecutive integers with equal sum of
squares. Mathematics Magazine. 1962; 35(3):155–64.
5. Guo S, Pan H, Sun Z-W. Mixed sums of squares and
­triangular numbers. Electronic Journal of Combinatorics
and Number Theory. 2007; A56:1–5.
6. Sun ZW. Sums of squares and triangular numbers. Acta
Arith. 2007; 127(2): 103–13.
7. Oh B-K, Sun Z-W. Mixed sums of squares and ­triangular
­numbers III. Journal of Number Theory. 2009;
129(4):964–9.
Vol 8 (15) | July 2015 | www.indjst.org
8. Farkas HM. Sums of squares and triangular numbers. Online
Journal of Analytic Combinatorics. 2006; 1(1):1–11.
9. Cilleruelo J, Luca F, Rue J, Zumalacarregui A. On the sum of
digits of some Sequences of integers. Cent Eur J Math. 2013;
11(1):188–95.
10. Allouche JP, Shallit JO. Sums of digits, overlaps and
­palindromes. Discrete Math Theor Comput Sci. 2000;
4(1):1–10.
11. Morgenbesser J. Gelfond’s sum of digits problems [Doctoral
thesis]. Vienna University of Technology; 1982.
12. Morgenbesser J. The sum of digits of squares in Z[i]. J
Number Theor. 2010; 130(7):1433–69.
13. Hickerson D, Kleber M. Reducing a set of subtracting
squares. Journal of Integer Sequences. 199; 2(2):10.
14. Liu J, Liu M C, Zhan T. Squares of primes and powers of 2.
J Number Theor . 2002; 92(1):99–116.
15. Browkin J, Brzeziñski J. On sequences of squares with
constant second differences. Can Math Bull. 2006;
49(4):481–91.
16. Ahmad A, Al-Busaidi SS, Awadalla M, Rizvi MAK,
Mohanan M. Computing and listing of number of possible m-­sequence generators of order n. Indian Journal of
Science and Technology. 2013; 6(10):5359–69.
17. Adam Grabowski. On square-free numbers. Formalized
Mathematics. 2013; 21(2):153–62.
18. Latushkin YA, Ushakov VN. On the representation of
Fibonacci and Lucas numbers as the sum of three squares.
Math Notes. 2012; 91(5):663–70.
19. Ahmad A, Al-Busaidi SS, Al-Musharafi MJ. On properties
of PIN sequences generated by LFSR - A generalized study
and simulation modeling. Indian Journal of Science and
Technology 2013; 6(10):5351–8.
20. Karthik GM, Pujeri RV. Constraint based periodic pattern
mining in multiple longest common subsequences. Indian
Journal of Science and Technology. 2013; 6(8):5047–57.
21. Spector L. The Math Page: Appendix Two. 2014.
22. Kaskin R, Karaath O. Some new properties of ­balancing
numbers and squared triangular numbers. Journal of
Integer Sequences. 2012; 15(1):1–13.
23. Wikipedia Square Number 24. Cohn JHE. Square Fibonacci
etc. Fibonacci Quartely. 1964; 2(2):109–13.
Indian Journal of Science and Technology
7